Uniqueness of Highly Representative Surface Embeddings

P. D. Seymour, Robin Thomas

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


Let Σ be a (connected) surface of "complexity" κ; that is, Σ may be obtained from a sphere by adding either 1/2κ handles or κ crosscaps. Let ρ ≥ 0 be an integer, and let Γ be a "ρ-representative drawing" in Σ; that is, a drawing of a graph in Σ so that every simple closed curve in Σ that meets the drawing in <ρ points bounds a disc in Σ. Now let Γ′ be another drawing, in another surface Σ′ of complexity κ′, so that Γ and Γ′ are isomorphic as abstract graphs. We prove that (i) If ρ ≥ 100 log κ/ log log κ (or ρ ≥ 100 if κ ≤ 2) then κ′ ≤ κ, and if κ′ = κ and Γ is simple and 3-connected there is a homeomorphism from Σ to Σ′ taking Γ to Γ′, (ii) if Γ is simple and 3-connected and Γ′ is 3-representative, and ρ ≥ (320, 5 log κ), then either there is a homeomorphism from Σ to Σ′ taking Γ to Γ′, or κ′ ≥ κ + 10-4 ρ2.

Original languageEnglish (US)
Pages (from-to)337-349
Number of pages13
JournalJournal of Graph Theory
Issue number4
StatePublished - Dec 1996
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


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